# Eigenvalues of an integral operator with non-degenerate kernel

I am trying to find the eigenvalues and eigenfunctions of the integral operator $Ku=\int_0^\pi k(x,y)u(y)dy$ with the following kernel: $k( x,y) = \sum\limits_{n=1}^\infty \frac{1}{n^2} \sin\big((n+1)x\big)\sin(ny)$.

Using the DCT we can exchange the sum and the integral to get: $$\sum\limits_{n=1}^\infty \frac{1}{n^2} \sin\big((n+1)x\big) \int_0^{\pi}\sin(ny)u(y)dy=\lambda u(y)$$

Now the LHS looks like a Fourier series of a function but I cannot guess which one. Any hints?

P.S. There is exactly the same question here Find eigenfunctions of the integral operator with kernel $\sum\limits_{n=1}^\infty \frac{1}{n^2} \sin((n+1)x)\sin(ny)$, with no answer and I followed the given hint there by replacing $u$ with its Fourier series but the integral terms argued there to be zero are definitely wrong.

The functions $\{ \sin(nx) \}_{n=1}^{\infty}$ form a complete orthogonal basis for $L^2(0,\pi)$ because they are the eigenfunction solutions of $$-y'' = \lambda y,\;\;\; y(0)=y(\pi)=0.$$ The normalization constants are $$\int_{0}^{\pi}\sin^2(nx)dx=\frac{1}{2}\int_{0}^{2\pi}\sin^2(nx)dx = \frac{1}{4}\int_{0}^{2\pi}\sin^2(nx)+\cos^2(nx)dx = \frac{\pi}{2}.$$ Any $f\in L^2(0,\pi)$ can be written uniquely as $\sum_{n=1}^{\infty}f_n\sin(nx)$ where $\{ f_n \}_{n=1}^{\infty} \in \ell^2$. In fact, $$f_n = \frac{2}{\pi}\int_{0}^{\pi}f(x)\sin(nx)dx,\;\; n=1,2,3,\cdots.$$ So your problem can be reduced to a problem on $\ell^2(\mathbb{N})$, where $u$ is represented by $\{ u_n \}_{n=1}^{\infty}$. The eigenfunction problem becomes a coefficient problem in $\ell^2$: $$Ku=\lambda u \\ \int_{0}^{\pi}u(y)k(x,y)dy=\lambda u(x) \\ \int_{0}^{\pi}u(y)\sum_{n=1}^{\infty}\frac{1}{n^2}\sin((n+1)x)\sin(ny)dy=\lambda u(x)$$ This gives coefficient equations after rewriting as \begin{align} \sum_{n=1}^{\infty}\frac{1}{n^2}\sin((n+1)x)\frac{\pi}{2}u_n&=\sum_{n=1}^{\infty}\lambda u_n\sin(nx) \\ \sum_{n=2}^{\infty}\frac{1}{(n-1)^2}\sin(nx)\frac{\pi}{2}u_{n-1} & = \sum_{n=1}^\infty \lambda u_n \sin(nx). \end{align} There is no $\sin(x)$ term on the left, which forces $\lambda u_1=0$. The general $u_n$ must satisfy $$\frac{\pi}{2}\frac{1}{(n-1)^2}u_{n-1}=\lambda u_n,\;\; n \ge 2.$$ If $\lambda \ne 0$, then $u_1 =0$ and every $u_n=0$ for $n > 1$ by the above. If $\lambda = 0$, then every $u_n=0$ by the above. So there are no eigenvalues of this operataor. That means that the operator $K$ is quasinilpotent with spectrum $\{0\}$.